# Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### Irrational numbers Cardinality.

The real numbers, $\mathbb{R}$, are uncountable and the rational numbers, $\mathbb{Q}$, are countable. We can write $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$. Since $\mathbb{Q}$ ...
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### Find the cardinality of $A \cup B$ [closed]

Let the sets $A=\{\frac{1}{1\times 2} , \frac{1}{2\times 3}, ... , \frac{1}{2021\times 2022}\}, B=\{ \frac{1}{2\times 4}, \frac{1}{3\times 5}, ..., \frac{1}{2020\times 2022}\}$. Find the cardinality ...
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### For infinite cardinals $\kappa$, we have $\kappa \otimes \kappa = \kappa$.

I am aware that other questions are quite similar to this; however, it seems like the other questions regarding the same statement are looking at proofs that seem somewhat different from the one I am ...
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### Cardinality of a set of disjoint open sub intervals of $( 0 ,1)$

Let $A$ be any collection of disjoint open subintervals of $(0 ,1)$ . Then what is maximum cardinality of $A$ ? I know one easy way to prove its countable is that every open interval has rational ...
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### proving the set of natural numbers is infinite (Tao Ex 2.6.3)

Tao's Analysis I 4th ed has the following exercise 3.6.3: Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
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### Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$

Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational numbers. $f$ is strictly increasing in both arguments. Can $f$ be one-to-one? This question is related to many ...
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### Can a countable union of subgroups of uncountable index in G be equal to G? [closed]

Let G be a group and $\{H_i\}_{i<\omega}$ be a countable family of subgroups of $G$, each of them of uncountable index. Can $G=\bigcup_{i<\omega} H_i$?
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### Is the cardinality of $\varnothing$ undefined?

It is intuitive that the cardinality of the empty set is $0$. However we are asked to demonstrate this using given definitions/axioms in Tao Analysis I 4th ed ex 3.6.2. My question arises as I think ...
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### How does measure theory deal with higher cardinalities?

The second part of the definition of a sigma-algebra is that countable unions of measurable sets are measurable. The second property of a measure is that the measure of countable unions of measurable ...
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### On the Singular Cardinal Hypothesis

I'm trying to find the proof of this result. If for each $\lambda\geq2^\omega$, $\lambda^\omega\le\lambda^+$, then the SCH holds. I'm not sure where to look. So if you have any info about this, please ...
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### Why is $\{0,1\}^{\Bbb N}$ uncountable? [duplicate]

In the book Measure and Integral : An introduction to real analysis, in chapter 8 Lp spaces, theorem 8.18, the authors give a counterexample to show that $l^\infty$ is not separable: Question: why is ...
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### Cardinal sum of powers

I'm trying to solve this exercise. Can anybody please help me: If $\kappa$ ist an infinite cardinal number with $cf(\kappa) = \kappa$ and for all $\mu < \kappa$ the inequality $2^{\mu} \leq \kappa$ ...
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### For all cardinals $\kappa, \lambda$ with $\lambda \geq cf(\kappa)$ the inequality $\kappa^{\lambda} > \kappa$ holds [duplicate]

I genuinely have no idea why the proposition in the title holds or how to show it. I am kind of new to cardinals and ordinals and very confused. If someone could explain, I would really appreciate ...
1 vote
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### Cardinal power towers

I am not an expert on large cardinals. I could not find any reference (and terminology) for the following question: We start with \lambda:=\aleph_0 \text{ [tet] } \omega = \aleph_0 ^ {\aleph_0 ^ {\...
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### Would well-founded Scott cardinals work in ZCA + Ranks?

Does original Zermelo's set theory + Regularity + Ranks, prove that every set is of equal size to some element of a Scott cardinal? The original Zermelo does include an axiom of Choice, and it admits ...
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### The cardinality of specific set $A\subset \mathbb{N}^{\mathbb{N}}$

Let $A$ be a set of total functions from the naturals to the naturals  such that for every $f\in A$ there is a finite set $B_f\subset \mathbb{N}$ , such that for every $x\notin B_f$ , $f(x+1)=f(x)+1$. ...
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### Induction does not preserve ordering between cardinality of sets?

Consider building a binary tree and consider it as a collection of points and edges. Here is one with five levels, numbered level $1$ at the top with $1$ node to level $5$ at the bottom with $16$ ...
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### A family of $\kappa^{<\omega}$ such that for every member in $\kappa$ is contained by all but finitely many elements of the family

Suppose that $\kappa$ is an uncountable cardinal. Let $\kappa^{<\omega}$ denote the family of all finite subsets of $\kappa$. Does there exist a family $S\subset\kappa^{<\omega}$ such that for ...
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### How should I should prove $\mathbb{R}\sim\{0,1\}^{ \mathbb{N}}$ [duplicate]

I've seen some argument about the binary representation, but I think it is not accurate because under some extreme cases, the rounding or bit constraint would results distinct reals also have the same ...
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### Dependence of the equation $a+1=a$ for infinite cardinals $a$ on the axiom of choice

let $A$ be a set such that for all $n \in$ N $A ≉ N_n$ where $N_n = \{ 0 ,1 ,2 ...... n-1\}$ and $a$ be the Cardinality of $A$ meaning ($|A| = a$) is it possible to prove that $a+1=a$ without ...
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### The cardinality of a convergent series [duplicate]

$A\subseteq \mathbb{R}^{+}$  is a set of positive real numbers ($0\notin \mathbb{R}^{+}$), for which there exists a positive real number $x$, such that for every finite subset $S\subseteq A$, the sum ...
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### Singular cardinals and $\kappa$-Lindelöf spaces

Say a space is $\kappa$-Lindelöf provided that for every open cover of the space, there exists a subcover of cardinality $<\kappa$. So $\aleph_0$-Lindelöf is compact, and $\aleph_1$-Lindelöf is ...
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### Are there any theorems that use the uncountability of the reals in their proof?

Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
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### Prove $C ∼ P(P(\mathbb{N}))$ when $C$ is defined as the set of all $S$ s.t $(z − m, z + m) ∩ S = ∅$ for every $z∈\mathbb{Z}$, $m∈\mathbb{R}$

First, I know that there is a very similar question - The cardinality of all sets $A$ such that $\forall \ z\in \mathbb{Z} \ , (z-k,z+k)\ \cap A=\emptyset : 0<k<0.5$ , but here I want to the ...
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### Why does countability misbehave in intuitionistic logic

On page 3 of this paper https://arxiv.org/pdf/2404.01256.pdf I spotted the claim: Definitions of countability in terms of injection into ℕ misbehave intuitionistically, because a subset of a ...
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### Is there a way to construct larger cardinals without choice axiom?

From Cantor's Theorem, we know that $|\mathcal{P}(X)| > |X|$. So, we can define inductively a set with cardinality $\aleph_n, \forall n \in \mathbb{N}$. Let $\lbrace A_i\rbrace_{i \in \mathbb{N}}$ ...
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### Infinite wacky race

Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the ...
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